Slide 1

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 2

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 3

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 4

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 5

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 6

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 7

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 8

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 9

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 10

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 11

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 12

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 13

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 14

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 15

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 16

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 17

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 18

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 19

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 20

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 21

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 22

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 23

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 24

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 25

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 26

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 27

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 28

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 29

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 30

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 31

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 32

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 33

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 34

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 35

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 36

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 37

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 38

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 39

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 40

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 41

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 42

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 43

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 44

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 45

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 46

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 47

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 48

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 49

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 50

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 51

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 52

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion


Slide 53

Exploratory Experimentation and Computation
Fields and IRMACS Workshop on
Discovery and Experimentation in Number Theory

(IRMACS, September 23, 2008)
Jonathan Borwein, FRSC
www.carma.newcastle.edu.au/~jb616
Director CARMA (Computer Assisted Research Mathematics and Applications)
Laureate Professor University of Newcastle, NSW
“[I]ntuition comes to us much earlier and with much less outside influence than formal
arguments which we cannot really understand unless we have reached a relatively
high level of logical experience and sophistication.”
…
“In the first place, the beginner must be convinced that proofs deserve to be studied,
George Polya (1887-1985)
that they have a purpose, that they are interesting.”

Revised
30/08/2008

Revised 23/09/09

Where I now live
wine

home

ABSTRACT
Jonathan M. Borwein
Newcastle
Abstract: The mathematical research community is facing a great
challenge to re-evaluate the role of proof in light of the growing power of
current computer systems, of modern mathematical computing packages,
and of the growing capacity to data-mine on the Internet. Add to that the
enormous complexity of many modern capstone results such as the
Poincaré conjecture, Fermat's last theorem, and the Classification of finite
simple groups. As the need and prospects for inductive mathematics
blossom, the requirement to ensure the role of proof is properly founded
remains undiminished. I shall look at the philosophical context and then
offer five bench-marking examples of the opportunities and challenges we
face, along with some interactive demonstrations. (Related paper)
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition,
and there was never any other object for it.” – Jacques Hadamard (1865-1963)

CARMA

NEWCASTLE
RESEARCH CENTRE
(9 core members)

OUTLINE
I. Working Definitions of:





Discovery
Proof (and of Mathematics)
Digital-Assistance
Experimentation (in Maths and in Science)

II. Five Core Examples:






p(n)
¼
Á(n)
³(3)
1/¼

“Keynes distrusted intellectual rigour of the Ricardian
type as likely to get in the way of original thinking and
saw that it was not uncommon to hit on a valid
conclusion before finding a logical path to it.”
- Sir Alec Cairncross, 1996

III. A Cautionary Finale
IV. Making Some Tacit Conclusions Explicit
“Mathematical proofs like diamonds should be hard
and clear, and will be touched with nothing but strict
reasoning.” - John Locke

“The Crucible”

PART I. PHILOSOPHY, PSYCHOLOGY, ETC
“This is the essence of science. Even though I do not understand
quantum mechanics or the nerve cell membrane, I trust those
who do. Most scientists are quite ignorant about most sciences
but all use a shared grammar that allows them to recognize
their craft when they see it. The motto of the Royal Society of
London is 'Nullius in verba' : trust not in words. Observation
and experiment are what count, not opinion and introspection.
Few working scientists have much respect for those who try to
interpret nature in metaphysical terms. For most wearers of
white coats, philosophy is to science as pornography is to
sex: it is cheaper, easier, and some people seem,
bafflingly, to prefer it. Outside of psychology it plays almost
no part in the functions of the research machine.” - Steve
Jones
 From his 1997 NYT BR review of Steve Pinker’s How the Mind Works.

WHAT is a DISCOVERY?
“discovering a truth has three components. First, there
is the independence requirement, which is just that one
comes to believe the proposition concerned by one’s
own lights, without reading it or being told. Secondly,
there is the requirement that one comes to believe it in
a reliable way. Finally, there is the requirement that
one’s coming to believe it involves no violation of one’s
epistemic state. …
In short , discovering a truth is coming to believe it
in an independent, reliable, and rational way.”
Marcus Giaquinto, Visual Thinking in Mathematics.
An Epistemological Study, p. 50, OUP 2007
“All truths are easy to understand once they are discovered; the point is
to discover them.” – Galileo Galilei

Galileo was not alone in this view
“I will send you the proofs of the theorems in this book. Since,
as I said, I know that you are diligent, an excellent teacher of
philosophy, and greatly interested in any mathematical
investigations that may come your way, I thought it might be
appropriate to write down and set forth for you in this same
book a certain special method, by means of which you will
be enabled to recognize certain mathematical questions with
the aid of mechanics. I am convinced that this is no less
useful for finding proofs of these same theorems.
For some things, which first became clear to me by the
mechanical method, were afterwards proved geometrically,
because their investigation by the said method does not
furnish an actual demonstration. For it is easier to supply
the proof when we have previously acquired, by the
method, some knowledge of the questions than it is to
find it without any previous knowledge.” - Archimedes
(287-212 BCE)
Archimedes to Eratosthenes in the introduction to The Method in
Mario Livio’s, Is God a Mathematician? Simon and Schuster, 2009

A Very Recent Discovery

(“independent, reliable and rational”)

W3(s)

4
W 1 ( 1) = 1
W 2 ( 1) =
¼
µ
¶
µ ¶
3 2 1=3 6 1
27 2 2=3 6 2
?
W 3 ( 1) =
¡
+
¡
: ( 1)
4
4
16 ¼
3
4 ¼
3
( 1) has been checked t o 170 places on 256
cores in about 15 m inut es. It orginat es w it h
our pro of ( JM B -Nuyens-St raub-W an) t hat f or
k = 0; 1; 2; 3; : : :
counts abelian squares
Ã1
!
Ã1
!
1
1
; ¡ k; ¡ k
; ¡ 2; ¡ 2
?
2
2
W 3 ( 2k) = 3 F 2
j4 and W 3 ( 1) = Re3 F 2
j4

1; 1

1; 1

WHAT is MATHEMATICS?
MATHEMATICS, n. a group of related subjects, including algebra,

geometry, trigonometry and calculus, concerned with the study
of number, quantity, shape, and space, and their interrelationships, applications, generalizations and abstractions.
 This definition, from my Collins Dictionary has no mention of proof, nor the
means of reasoning to be allowed (vidé Giaquinto). Webster's contrasts:
INDUCTION, n. any form of reasoning in which the conclusion,

though supported by the premises, does not follow from them
necessarily.
and
DEDUCTION, n. a. a process of reasoning in which a conclusion

follows necessarily from the premises presented, so that the
conclusion cannot be false if the premises are true.
b. a conclusion reached by this process.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics just the
same as in physics.” - Kurt Gödel (in his 1951 Gibbs Lecture)
echoes of Quine

WHAT is a PROOF?
“PROOF, n. a sequence of statements, each of which is either
validly derived from those preceding it or is an axiom or
assumption, and the final member of which, the conclusion , is
the statement of which the truth is thereby established. A direct
proof proceeds linearly from premises to conclusion; an indirect
proof (also called reductio ad absurdum) assumes the falsehood
of the desired conclusion and shows that to be impossible. See
also induction, deduction, valid.”
Borowski & JB, Collins Dictionary of Mathematics
INDUCTION, n. 3. ( Logic) a process of reasoning in which a general conclusion is drawn from a
set of particular premises, often drawn from experience or from experimental evidence. The
conclusion goes beyond the information contained in the premises and does not follow
necessarily from them. Thus an inductive argument may be highly probable yet lead to a
false conclusion; for example, large numbers of sightings at widely varying times and
places provide very strong grounds for the falsehood that all swans are white.

“No. I have been teaching it all my life, and I do not want to have my ideas
upset.” - Isaac Todhunter (1820-1884) recording Maxwell’s response when asked
whether he would like to see an experimental demonstration of conical refraction.

Decide for yourself

WHAT is DIGITAL ASSISTANCE?
 Use of Modern Mathematical Computer Packages



Symbolic, Numeric, Geometric, Graphical, …

 Use of More Specialist Packages or General Purpose Languages


Fortran, C++, CPLEX, GAP, PARI, MAGMA, …

 Use of Web Applications


Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer,
Euclid in Java, Weeks’ Topological Games, …

 Use of Web Databases


Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math,
DLMF, MacTutor, Amazon, …, Wolfram Alpha (??)

 All entail data-mining [“exploratory experimentation” and “widening
technology” as in pharmacology, astrophysics, biotech… (Franklin)]



Clearly the boundaries are blurred and getting blurrier
Judgments of a given source’s quality vary and are context dependent

“Knowing things is very 20th century. You just need to be able to find
things.”- Danny Hillis on how Google has already changed how we think in
Achenblog, July 1 2008
- changing cognitive styles

Exploratory Experimentation
Franklin argues that Steinle's “exploratory experimentation” facilitated
by “widening technology”, as in pharmacology, astrophysics, medicine,
and biotechnology, is leading to a reassessment of what legitimates
experiment; in that a “local model" is not now prerequisite. Hendrik
Sørenson cogently makes the case that experimental mathematics (as
‘defined’ below) is following similar tracks:

“These aspects of exploratory experimentation and wide
instrumentation originate from the philosophy of (natural)
science and have not been much developed in the context
of experimental mathematics. However, I claim that e.g.
the importance of wide instrumentation for an exploratory
approach to experiments that includes concept formation
is also pertinent to mathematics.”
In consequence, boundaries between mathematics and the natural
sciences and between inductive and deductive reasoning are blurred
and getting more so.

Changing User Experience and Expectations
What is attention? (Stroop test, 1935)
1. Say the color represented
by the word.
2. Say the color represented
by the font color.
High (young) multitaskers perform
#2 very easily. They are great at
suppressing information.
http://www.snre.umich.edu/eplab/demos/st0/stroop_program/stroopgraphicnonshockwave.gif

Acknowledgements: Cliff Nass, CHIME lab, Stanford (interference and twitter?)

Other Cognitive Shifts

Harwell 1951-1973

Science Online August 13, 2009

 Potentially hostile to mathematical research patterns

Experimental

Mathodology

1. Gaining insight and intuition

Science News
2004

2. Discovering new relationships

“Computers are
3. Visualizing
math
principles
useless,
they
can
only give
answers.”
4. Testing
and especially
falsifying
conjectures
Pablo Picasso

Experimental Mathodology

5. Exploring a possible result to see
if it merits formal proof
6. Suggesting approaches for
formal proof
7. Computing replacing lengthy
hand derivations

8. Confirming analytically derived
results

Comparing –y2ln(y) (red) to y-y2 and y2-y4

Example 0. What is that number? (1995-2008)
In I995 or so Andrew Granville emailed me the number

and challenged me to identify it (our inverse calculator was new in
those days).
Changing representations, I asked for its continued fraction? It was

I reached for a good book on continued fractions and found the answer

where I0 and I1 are Bessel functions of the first kind. (Actually I knew
that all arithmetic continued fractions arise in such fashion).
In 2009 there are at least three other strategies:
• Given (1), type “arithmetic progression”, “continued fraction” into Google
• Type “1,4,3,3,1,2,7,4,2” into Sloane’s Encyclopaedia of Integer Sequences
I illustrate the results on the next two slides:

“arithmetic progression”, “continued fraction”
In Google on October 15 2008 the first three hits were
Continued Fraction Constant -- from Wolfram MathWorld
- 3 visits - 14/09/07Perron (1954-57) discusses continued fractions having
terms even more general than the arithmetic progression and relates
them to various special functions. ...
mathworld.wolfram.com/ContinuedFractionConstant.html - 31k
HAKMEM -- CONTINUED FRACTIONS -- DRAFT, NOT YET PROOFED
The value of a continued fraction with partial quotients increasing in
arithmetic progression is I (2/D) A/D [A+D, A+2D, A+3D, . ...
www.inwap.com/pdp10/hbaker/hakmem/cf.html - 25k On simple continued fractions with partial quotients in arithmetic ...
0. This means that the sequence of partial quotients of the continued
fractions under. investigation consists of finitely many arithmetic
progressions (with ...
www.springerlink.com/index/C0VXH713662G1815.pdf - by P Bundschuh
– 1998
Moreover the MathWorld entry includes

In the Integer Sequence Data Base
The Inverse Calculator
returns

Best guess:
BesI(0,2)/BesI(1,2)
• We show the ISC on
another number next
• Most functionality of
ISC is built into “identify”
in Maple.

• There’s also Wolfram
®
“The price of metaphor is eternal vigilance.” - Arturo Rosenblueth & Norbert Wiener
quoted by R. C. Leowontin, Science p.1264, Feb 16, 2001 [Human Genome Issue].

The ISC in Action

• ISC+ runs on Glooscap
• Less lookup & more
algorithms than 1995

Computer Algebra + Interactive Geometry:
the Grief is in the GUI

Numerical errors
in using double
precision

Stability using
Maple input

This picture is worth 100,000 ENIACs

ENIAC

Eckert & Mauchly (1946)

The number of ENIACS
needed to store the 20Mb
TIF file the Smithsonian
sold me

Projected Performance

PART II MATHEMATICS
“The question of the ultimate foundations and the
ultimate meaning of mathematics remains open: we
do not know in what direction it will find its final
solution or even whether a final objective answer can
be expected at all. 'Mathematizing' may well be a
creative activity of man, like language or music, of
primary originality, whose historical decisions defy
complete objective rationalisation.” - Hermann Weyl

In “Obituary: David Hilbert 1862 – 1943,” RSBIOS, 4, 1944, pp. 547-553;
and American Philosophical Society Year Book, 1944, pp. 387-395, p. 392.

IIa. The Partition Function (1991-2009)
Consider the number of additive partitions, p(n), of n. Now
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5)=7. The ordinary generating function discovered by Euler is
X1
Y1 ¡
¢
n
k ¡ 1
p( n) q =
1¡ q
:
( 1)
n= 0

k= 1

(Use the geometric formula for 1/(1-qk) and observe how powers of qn occur.)

The famous computation by MacMahon of p(200)=3972999029388 done
symbolically and entirely naively using (1) on an Apple laptop took 20 min
in 1991, and about 0.17 seconds in 2009. Now it took 2 min for
p(2000) = 4720819175619413888601432406799959512200344166
In 2008, Crandall found p(109) in 3 seconds on a laptop, using the HardyRamanujan-Rademacher ‘finite’ series for p(n) with FFT methods.
Such fast partition-number evaluation let Crandall find probable primes
p(1000046356) and p(1000007396). Each has roughly 35,000 digits.
When does easy access to computation discourages innovation: would Hardy
and Ramanujan have still discovered their marvellous formula for p(n)?

Cartoon

IIb. The computation of Pi (1986-2009)
BB4: Pi to 1.7
trillion places
in 20 steps

A random walk on a
million digits of Pi

Moore’ s Law Marches On
1986: It took Bailey 28 hours to compute 29.36 million digits
on 1 cpu of the then new CRAY-2 at NASA Ames using (BB4).
Confirmation using another BB quadratic algorithm took 40
hours.
This uncovered hardware and software errors on the CRAY.

2009 Takahashi on 1024 cores of a 2592 core Appro Xtreme X3 system 1.649 trillion digits via (Salamin-Brent) took 64
hours 14 minutes with 6732 GB of main memory, and (BB4)
took 73 hours 28 minutes with 6348 GB of main memory.
The two computations differed only in the last 139 places.
“The most important aspect in solving a mathematical problem is the conviction
of what is the true result. Then it took 2 or 3 years using the techniques that had
been developed during the past 20 years or so.” - Leonard Carleson (Lusin’s
problem on p.w. convergence of Fourier series in Hilbert space)

Cartoon

II c. Guiga and Lehmer (1932-2009)
As another measure of what changes over time and what doesn't,
consider two conjectures regarding Euler’s totient Á(n) which
counts positive numbers less than and relatively prime to n.
Giuga's conjecture (1950) n is prime if and only if
Gn :=

n¡
X 1

k n¡ 1 ´ ( n ¡ 1) m od n:

k= 1

Counterexamples are Carmichael numbers (rare birds only
proven infinite in 1994) and more: if a number n = p1 pm
with m>1 prime factors pi is a counterexample to Giuga's
conjecture then the primes are distinct and satisfy
Xm 1
i = 1 pi

> 1

and they form a normal sequence: pi 1 mod pj
(3 rules out 7, 13,... and 5 rules out 11, 31, 41,...)

Guiga’s Conjecture (1951-2009)
With predictive experimentally-discovered heuristics, we built an
efficient algorithm to show (in several months in 1995) that any
counterexample had 3459 prime factors and so exceeded
1013886  1014164 in a 5 day desktop 2002 computation.
The method fails after 8135 primes---my goal is to exhaust it.

2009 While preparing this talk, I obtained almost as good a
bound of 3050 primes in under 110 minutes on my notebook
and a bound of 3486 primes in 14 hours: using Maple not as
before C++ which being compiled is faster but in which the
coding is much more arduous.
One core of an eight-core MacPro obtained 3592 primes and
1016673 digits in 13.5 hrs in Maple. (Now running on 8 cores.)

Lehmer’s Conjecture (1932-2009)
A tougher related conjecture is
Lehmer's conjecture (1932) n is prime if and only if
Á( n) j( n ¡ 1)
He called this “as hard as the existence of odd perfect numbers.”
Again, prime factors of counterexamples form a normal sequence,
but now there is little extra structure.

In a 1997 SFU M.Sc. Erick Wong verified this for 14 primes,
using normality and a mix of PARI, C++ and Maple to press the
bounds of the ‘curse of exponentiality.’
The related Á(n) |(n+1) is has 8 solutions with at most 7 factors (6 factors is due
n
to Lehmer).
Recall Fn:=22 +1 the Fermat primes. The solutions are 2, 3,
3.5, 3.5.17, 3.5.17.257, 3.5.17.257.65537 and a rogue pair: 4919055 and
6992962672132095, but 8 factors seems out of sight.
Lehmer “couldn’t” factor 6992962672132097= 73£95794009207289. If prime,
a 9th would exist: Á(n) |(n+1) and n+2 prime ) N:=n(n+2) satisfies Á(N)|(N+1)

Cartoon

II d. Apéry-Like Summations
The following formulas for (n) have been known for many decades:

The RH in Maple

These results have led many to speculate that

might be some nice rational or algebraic value.
Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial
with degree at most 25, then at least one coefficient has 380 digits.
"He (Gauss) is like the fox, who effaces his tracks in the sand
with his tail.“ - Niels Abel (1802-1829)

Two more things about ³(5)

Nothing New under the Sun

 The case a=0 above is Apéry’s formula for (3) !
Andrei Andreyevich Markov
(1856-1922)

Two Discoveries: 1995 and 2005
 Two computer-discovered generating functions
 (1) was ‘intuited’ by Paul Erdös (1913-1996)

 and (2) was a designed experiment
 was proved by the computer (Wilf-Zeilberger)
 and then by people (Wilf included)
 What about 4k+1?

x=0 gives (b) and (a) respectively

Euler
(1707-73)

Apéry summary
1. via PSLQ to
5,000
digits
Riemann
(120
terms)
(1826-66)

1

2005 Bailey, Bradley
& JMB discovered and
proved - in 3Ms - three
equivalent binomial
identities

2

2. reduced
as hoped

3

3. was easily computer proven (WilfZeilberger) (now 2 human proofs)

Cartoon

II e: Ramanujan-Like Identities

A lit t le lat er D avid and Gregory Chudnovsky
p
f ound t he f ollow ing variant , w hich lies in Q( ¡ 163)
p
rat her t han Q( 58) :
1
=
¼

12

X1
k= 0

( ¡ 1) k ( 6k) ! ( 13591409 + 545140134k)
( 3k) ! ( k!) 3 640320 3k+ 3=2

:

( 2)

E ach t erm of ( 2) adds 14 correct digit s.
The brothers used (2) several times --- culminating in a 1994 calculation to over four billion
decimal digits. Their remarkable story was told in a Pulitzer-winning New Yorker article.

New Ramanujan-Like Identities
Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two using the Wilf-Zeilberger algorithm. He
ascribed the third to Gourevich, who found it using integer relation methods.
It is true but has no proof.
As far as we can tell there are no higher-order analogues!

Example of Use of Wilf-Zeilberger, I
The first two recent experimentally-discovered identities are

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Wilf-Zeilberger, II
When we define

Zeilberger's theorem gives the identity

which when written out is

http://ddrive.cs.dal.ca/~isc/portal

A limit argument and Carlson’s theorem completes the proof…

Searches for Additional Formulas
We have no PSLQ over number fields so we searched for
additional formulas of either the following forms:

where c is some linear combination of

where each of the coefficients pi is a linear combination of
and where  is chosen as one of the following:

Relations Found by PSLQ

(with Guillera’s three we found all known series and no more)

Baruah, Berndt, Chan, “Ramanujan Series for 1/¼: A Survey.” Aug 09, MAA Monthly

Cartoon

III. A Cautionary Example
These constants agree to 42 decimal digits accuracy, but
are NOT equal:

Computing this integral is (or was) nontrivial, due largely to
difficulty in evaluating the integrand function to high
precision.
Fourier analysis explains
this happens when a
hyperplane meets a
hypercube (LP) …

IV. Some Conclusions
 We like students of 2010 live in an information-rich, judgement-poor world
 The explosion of information is not going to diminish


nor is the desire (need?) to collaborate remotely

 So we have to learn and teach judgement (not obsession with plagiarism)
 that means mastering the sorts of tools I have illustrated
 We also have to acknowledge that most of our classes will contain a very
broad variety of skills and interests (few future mathematicians)
 properly balanced, discovery and proof can live side-by-side and
allow for the ordinary and the talented to flourish in their own fashion
 Impediments to the assimilation of the tools I have illustrated are myriad




as I am only too aware from recent experiences

These impediments include our own inertia and
 organizational and technical bottlenecks (IT - not so much dollars)
 under-prepared or mis-prepared colleagues
 the dearth of good modern syllabus material and research tools
 the lack of a compelling business model (societal goods)
“The plural of 'anecdote' is not 'evidence'.”
- Alan L. Leshner (Science's publisher)

Further Conclusions
New techniques now permit integrals,
infinite series sums and other entities
to be evaluated to high precision
(hundreds or thousands of digits), thus
permitting PSLQ-based schemes to
discover new identities.
These methods typically do not
suggest proofs, but often it is much
easier to find a proof (say via WZ)
when one “knows” the answer is right.

Full details of all the examples are in Mathematics by Experiment or its
companion volume Experimentation in Mathematics written with Roland
Girgensohn. A “Reader’s Digest” version of these is available at
www.experimentalmath.info along with much other material.
“Anyone who is not shocked by quantum theory has not understood a

single word.” - Niels Bohr

A Sad Story (UK)
1. Teaching Maths In 1970 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price.
What is his profit?
2. Teaching Maths In 1980 A logger sells a lorry load of timber
for £1000. His cost of production is 4/5 of the selling price, or
£800. What is his profit?
3. Teaching Maths In 1990 A logger sells a lorry load of timber
for £1000. His cost of production is £800. Did he make a
profit?
4. Teaching Maths In 2000 A logger sells a lorry load of timber
for £1000. His cost of production is £800 and his profit is £200.
Underline the number 200.
5. Teaching Maths In 2008 A logger cuts down a beautiful
forest because he is a totally selfish and inconsiderate bastard
and cares nothing for the habitat of animals or the
preservation of our woodlands. He does this so he can make
a profit of £200. What do you think of this way of making a
living?
Topic for class participation after answering the question: How did the birds and squirrels
feel as the logger cut down their homes? (There are no wrong answers. If you are upset
about the plight of the animals in question counselling will be available.)

More Self Promotion